Univalent Functions in Two-Dimensional Free Boundary Problems

The main goal of the paper is to bring together methods of the classical theory of univalent functions and some problems of fluid mechanics. Our interest centers on free boundary problems. We study the time evolution of the free boundary of a viscous fluid in the zero- and nonzero-surface-tension models for planar flows in Hele-Shaw cells either with an extending to infinity free boundary or with a bounded free boundary. We consider special classes of univalent functions that admit an explicit geometric interpretation to characterize the shape of the free interface. Another model is two-dimensional solidification/melting of a nucleus in a forced flow.